Kerodon

Corollary 4.4.3.20. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories, and let $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ be the fiber of $q$ over an object $D \in \operatorname{\mathcal{D}}$. Then the canonical map $(\operatorname{\mathcal{C}}_{D})^{\simeq } \rightarrow \{ D\} \times _{\operatorname{\mathcal{D}}^{\simeq }} \operatorname{\mathcal{C}}^{\simeq }$ is an isomorphism. In other words, the inclusion functor $\operatorname{\mathcal{C}}_{D} \hookrightarrow \operatorname{\mathcal{C}}$ is conservative.